Abstract
Let $(A,\mathfrak{m})$ be a noetherian local ring and $J$ an $\mathfrak{m}$-primary ideal. Elias [3] proved that $\operatorname{depth}(G(J^k))$ is constant for $k \gg 0$ and denoted this number by $\sigma(J)$. In this paper, we prove the non-positivity for the Hilbert coefficients $e_i(J)$ under some conditions for $\sigma(J)$. In case of $J = Q$ is a parameter ideal, we establish bounds for the Hilbert coefficients of $Q$ in terms of the dimension and the first Hilbert coefficient $e_1(Q)$.
Funding Statement
The paper was completed while the first author was visiting
the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the
VIASM for financial support and hospitality. This work was partially supported by the Core
Research Program of Hue University, Grant No. NCM.DHH.2020.15.
Acknowledgments
The authors thank the referees for his careful reading of the manuscript and his valuable hints.
Citation
Cao Huy Linh. Ton That Quoc Tan. "Bounds for Hilbert Coefficients." Taiwanese J. Math. 25 (6) 1159 - 1172, December, 2021. https://doi.org/10.11650/tjm/210602
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